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The Minimum of the Total Potential
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analysis applied arbitrary assumed beam column bending moments bending rigidity buckling load calculated Castigliano's second theorem caused complementary energy components compressive conditions of equilibrium Consequently constant corresponding cross section curve deflected shape deformations derivative diagram differential equation dummy load end load Euler load expression external loads given in Eq Hardy Cross Hence Hooke's law horizontal integral joint length magnitude mass points maximum member of Eq minimal principle modulus monocoque obtained plane positive principle of virtual problem quantities Rayleigh-Ritz method reaction redundant right-hand member rigid frame rotation safety factor Saint-Venant's principle shear center shear force shear stress shown in Fig simply supported solution space framework stability statically indeterminate strain energy stored stress-strain stress-strain curve structure Substitution symmetry tension coefficients theorem theory torsion total potential unknown vanish variation vertical virtual displacements warping group yield hinge yield stress Young's modulus